Differential geometry, lie groups, and symmetric spaces:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Mathematical Soc.
2001
|
| Edition: | Reprinted with corr. |
| Series: | Graduate studies in mathematics
34 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXVI, 641 S. |
| ISBN: | 0821828487 9780821828489 |
Staff View
MARC
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| 245 | 1 | 0 | |a Differential geometry, lie groups, and symmetric spaces |c Sigurdur Helgason |
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| 264 | 1 | |a Providence, RI |b American Mathematical Soc. |c 2001 | |
| 300 | |a XXVI, 641 S. | ||
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| 650 | 4 | |a Géométrie différentielle | |
| 650 | 4 | |a Lie, Groupes de | |
| 650 | 4 | |a Geometry, Differential | |
| 650 | 4 | |a Lie groups | |
| 650 | 4 | |a Symmetric spaces | |
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Record in the Search Index
| _version_ | 1804128819773177856 |
|---|---|
| adam_text | CONTENTS
Preface
................ xiii
Preface to the
2001
Printing
............ xvii
Suggestions to the Readeh
............ xix
Sequel to the Present Volume
............ xxi
Groups and Geometric Analysis Contents
.......... xxiii
Geometric Analysis on Symmetric Spaces Contents
........ xxv
CHAPTER I
Elementary Differential Geometry
1.
Manifolds
............... 2
2.
Tensor Fields
.............. 8
1.
Vector Fields and 1-Forms
.......... 8
2.
Tensor Algebra
............ 13
3.
The
Grassman
Algebra
........... 17
4.
Exterior Differentiation
........... 19
3.
Mappings
............... 22
1.
The Interpretation of the Jacobian
........ 22
2.
Transformation of Vector Fields
......... 24
3.
Effect on Differential Forms
.......... 25
4. Affine
Connections
............. 26
5.
Parallelism
............... 28
6.
The Exponential Mapping
........... 32
7.
Covariant Differentiation
........... 40
8.
The Structural Equations
........... 43
9.
The Riemannian Connection
........... 47
10.
Complete Riemannian Manifolds
.......... 55
11.
Isometries
............... 60
12.
Sectional Curvature
............. 64
13.
Riemannian Manifolds of Negative Curvature
....... 70
14.
Totally Geodesic Submanifolds
.......... 78
15.
Appendix
............... 82
Λ
Topology
.............. 82
2.
Mappings of Constant Rank
.......... 86
Exercises and Further Results
.......... 88
Notes
................ 95
CHAPTER II
Lie Groups and Lie Algebras
1.
The Exponential Mapping
........... 98
1.
The Lie Algebra of a Lie Group
......... 98
X
CONTENTS
2.
The Universal Enveloping Algebra
........ 100
3.
Left Invariant
Affine
Connections
. . . . . . . . 102
4.
Taylor s Formula and the Differential of the Exponential Mapping
. . 104
2.
Lie Subgroups and Subalgebras
.......... 112
3.
Lie Transformation Groups
. . . . . . . ■ · · .120
4.
Coset Spaces and Homogeneous Spaces
........ 123
5.
The Adjoint Group
............. 126
6. Semisimple
Lie Groups
. . . . . . . . . . · .131
7.
Invariant Differential Forms
........... 135
8.
Perspectives
.............. 144
Exercises and Further Results
.......... 147
Notes
................ 153
CHAPTER 111
Structure of
Semisimple
Lie Algebras
1.
Preliminaries
.............. 155
2.
Theorems of Lie and
Engel........... 158
3.
Cartan Subalgebras
............. 162
4.
Root Space Decomposition
........... 165
5.
Significance of the Root Pattern
.......... 171
6.
Real Forms
.............. 178
7.
Cartan Decompositions
............ 182
8.
Examples. The Complex Classical Lie Algebras
....... 186
Exercises and Further Results
.......... 191
Notes
................ 196
CHAPTER IV
Symmetric Spaces
1. Affine
Locally Symmetric Spaces
.......... 198
2.
Groups of Isometries
............ 201
3.
Riemannian Globally Symmetric Spaces
........ 205
4.
The Exponential Mapping and the Curvature
....... 214
5.
Locally and Globally Symmetric Spaces
........ 218
6.
Compact Lie Groups
............ 223
7.
Totally Geodesic Submanifolds. Lie Triple Systems
...... 224
Exercises and Further Results
.......... 226
Notes
................ 227
CHAPTER V
Decomposition of Symmetric Spaces
1.
Orthogonal Symmetric Lie Algebras
......... 229
2.
The Duality
.............. 235
3.
Sectional Curvature of Symmetric Spaces
........ 241
4.
Symmetric Spaces with
Semisimple
Groups of Isometries
..... 243
CONTENTS Xl
5. Notational Conventions............ 244
6.
Rank of
Symmetrie Spaces........... 245
Exercises and Further Results
.......... 249
Notes................ 251
CHAPTER VI
Symmetric
Spaces
of the
N
o neo
m
pact Type
1.
Decomposition of
a
Semisimple
Lie Group
........ 252
2.
Maximal Compact Subgroups and Their Conjugacy
...... 256
3.
The Iwasawa Decomposition
........... 257
4. Nilpotent
Lie Groups
............ 264
5.
Global Decompositions
............ 270
6.
The Complex Case
............. 273
Exercises and Further Results
.......... 275
Notes
................ 279
CHAPTER
VII
Symmetric Spaces of the Compact Type
1.
The Contrast between the Compact Type and the Noncompact Type
. .281
2.
The Weyl Group and the Restricted Roots
........ 283
3.
Conjugate Points. Singular Points. The Diagram
....... 293
4.
Applications to Compact Groups
.......... 297
5.
Control over the Singular Set
........... 303
6.
The Fundamental Group and the Center
........ 307
7.
The
Affine
Weyl Group
............ 314
8.
Application to the Symmetric Space UjK
........ 318
9.
Classification of Locally Isometric Spaces
........ 325
10.
Geometry of UjK. Symmetric Spaces of Rank One
...... 327
11.
Shortest Geodesies and Minimal Totally Geodesic Spheres
.... 334
12.
Appendix. Results from Dimension Theory
........ 344
Exercises and Further Results
.......... 347
Notes
................ 350
CHAPTER
VIII
Hermitian Symmetric Spaces
1.
Almost Complex Manifolds
........... 352
2.
Complex Tensor Fields. The
Rìcci
Curvature
. . . . . . .356
3.
Bounded Domains. The Kernel Function
........ 364
4.
Hermitian Symmetric Spaces of the Compact Type and the Noncompact Type
372
5.
Irreducible Orthogonal Symmetric Lie Algebras
....... 377
6.
Irreducible Hermitian Symmetric Spaces
........ 381
7.
Bounded Symmetric Domains
.......... 382
Exercises and Further Results
.......... 396
Notes
................ 400
XU CONTENTS
CHAPTER IX
Structure of
Semisimple
Lie Groups
1.
Cartan, Iwasawa, and Bruhat Decompositions
....... 401
2.
The Rank-One Reduction
........... 407
3.
The SU(2,
1)
Reduction
............ 409
4.
Cartan Subalgebras
............. 418
5.
Automorphisms
............. 421
6.
The Multiplicities
............. 428
7.
Jordan Decompositions
............ 430
Exercises and Further Results
.......... 434
Notes
................ 436
CHAPTER X
The Classification of Simple Lie Algebras
and of Symmetric Spaces
1.
Reduction of the Problem
........... 438
2.
The Classical Groups and Their Cartan Involutions
...... 444
1.
Some Matrix Groups and Their Lie Algebras
...... 444
2.
Connectivity Properties
........... 447
3.
The Jnvolutive Automorphisms of the Classical Compact Lie Algebras
. 451
З»
Root Systems
.............. 455
1.
Generalities
............. 455
2.
Reduced Root Systems
........... 459
3.
Classification of Reduced Root Systems. Coxeter Graphs and Dynkin
Diagrams
............. 461
4.
The Nonreduced Root Systems
......... 474
5.
The Highest Root
............ 475
6.
Outer Automorphisms and the Covering Index
...... 478
4.
The Classification of Simple Lie Algebras over
С
...... 481
5.
Automorphisms of Finite Order of
Semisimple
Lie Algebras
.... 490
6.
The Classifications
............. 515
1.
The Simple Lie Algebras over
С
and Their Compact Real Forms. The
Irreducible Riemannian Globally Symmetric Spaces of Type II and Type IV
515
2.
The Real Forms of Simple Lie Algebras over C. Irreducible Riemannian
Globally Symmetric Spaces of Type I and Type IV
..... 517
3.
Irreducible Hermitian Symmetric Spaces
....... 518
4.
Coincidences between Different Classes. Special Isomorphisms
. . . 518
Exercises and Further Results
.......... 520
Notes
................ 535
Solutions to Exercises
............. 538
Some Details
............... 586
Bibliography
............... 599
List of Notational Conventions
........... 629
Symbols Frequently Used
............ 632
Index
................. 635
Reviews for the First Edition
............ 641
|
| any_adam_object | 1 |
| author | Sigurður Helgason 1927-2023 |
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| dewey-full | 516.3/6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/6 |
| dewey-search | 516.3/6 |
| dewey-sort | 3516.3 16 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Reprinted with corr. |
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| id | DE-604.BV013968264 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:55:18Z |
| institution | BVB |
| isbn | 0821828487 9780821828489 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009560937 |
| oclc_num | 45917187 |
| open_access_boolean | |
| owner | DE-703 DE-355 DE-BY-UBR DE-91G DE-BY-TUM DE-83 DE-11 DE-19 DE-BY-UBM DE-20 |
| owner_facet | DE-703 DE-355 DE-BY-UBR DE-91G DE-BY-TUM DE-83 DE-11 DE-19 DE-BY-UBM DE-20 |
| physical | XXVI, 641 S. |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | American Mathematical Soc. |
| record_format | marc |
| series | Graduate studies in mathematics |
| series2 | Graduate studies in mathematics |
| spelling | Sigurður Helgason 1927-2023 Verfasser (DE-588)123045762 aut Differential geometry, lie groups, and symmetric spaces Sigurdur Helgason Reprinted with corr. Providence, RI American Mathematical Soc. 2001 XXVI, 641 S. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 34 Espaces symétriques GEOMETRIA DIFERENCIAL larpcal Géométrie différentielle Lie, Groupes de Geometry, Differential Lie groups Symmetric spaces Lie-Gruppe (DE-588)4035695-4 gnd rswk-swf Symmetrischer Raum (DE-588)4184206-6 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 s DE-604 Differentialgleichung (DE-588)4012249-9 s Symmetrischer Raum (DE-588)4184206-6 s Lie-Gruppe (DE-588)4035695-4 s Graduate studies in mathematics 34 (DE-604)BV009739289 34 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009560937&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Sigurður Helgason 1927-2023 Differential geometry, lie groups, and symmetric spaces Graduate studies in mathematics Espaces symétriques GEOMETRIA DIFERENCIAL larpcal Géométrie différentielle Lie, Groupes de Geometry, Differential Lie groups Symmetric spaces Lie-Gruppe (DE-588)4035695-4 gnd Symmetrischer Raum (DE-588)4184206-6 gnd Differentialgleichung (DE-588)4012249-9 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| subject_GND | (DE-588)4035695-4 (DE-588)4184206-6 (DE-588)4012249-9 (DE-588)4012248-7 |
| title | Differential geometry, lie groups, and symmetric spaces |
| title_auth | Differential geometry, lie groups, and symmetric spaces |
| title_exact_search | Differential geometry, lie groups, and symmetric spaces |
| title_full | Differential geometry, lie groups, and symmetric spaces Sigurdur Helgason |
| title_fullStr | Differential geometry, lie groups, and symmetric spaces Sigurdur Helgason |
| title_full_unstemmed | Differential geometry, lie groups, and symmetric spaces Sigurdur Helgason |
| title_short | Differential geometry, lie groups, and symmetric spaces |
| title_sort | differential geometry lie groups and symmetric spaces |
| topic | Espaces symétriques GEOMETRIA DIFERENCIAL larpcal Géométrie différentielle Lie, Groupes de Geometry, Differential Lie groups Symmetric spaces Lie-Gruppe (DE-588)4035695-4 gnd Symmetrischer Raum (DE-588)4184206-6 gnd Differentialgleichung (DE-588)4012249-9 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Espaces symétriques GEOMETRIA DIFERENCIAL Géométrie différentielle Lie, Groupes de Geometry, Differential Lie groups Symmetric spaces Lie-Gruppe Symmetrischer Raum Differentialgleichung Differentialgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009560937&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009739289 |
| work_keys_str_mv | AT sigurðurhelgason differentialgeometryliegroupsandsymmetricspaces |